CONFORMALLY HOMEOMORPHIC LORENTZ SURFACES NEED NOT BE CONFORMALLY DIFFEOMORPHIC

A Lorentz surface L is an ordered pair (S, [h]) where S is an oriented C-infinity 2-manifold and [h] the set of all C-infinity metrics confo rmally equivalent to a fixed C-infinity Lorentzian metric h on S. (Thu s Lorentz surfaces are the indefinite metric analogs of Riemann surfac es.) This paper describes subsets of the Minkowski 2-plane which are c onformally homeomorphic, but not even C-1 conformally diffeomorphic. I t also describes subsets of the Minkowski 2-plane which are C-j but no t C-j+1 conformally diffeomorphic for any fixed j = 1, 2,... . Finally , the paper describes a Lorentz surface conformally homeomorphic to a subset of the Minkowski 2-plane, but not C-1 conformally diffeomorphic to any subset of the Minkowski 2-plane.